L(s) = 1 | − 2-s − 0.665·3-s + 4-s + 0.670·5-s + 0.665·6-s − 3.39·7-s − 8-s − 2.55·9-s − 0.670·10-s − 2.66·11-s − 0.665·12-s + 2.25·13-s + 3.39·14-s − 0.445·15-s + 16-s + 3.18·17-s + 2.55·18-s + 4.73·19-s + 0.670·20-s + 2.26·21-s + 2.66·22-s + 23-s + 0.665·24-s − 4.55·25-s − 2.25·26-s + 3.69·27-s − 3.39·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.384·3-s + 0.5·4-s + 0.299·5-s + 0.271·6-s − 1.28·7-s − 0.353·8-s − 0.852·9-s − 0.211·10-s − 0.803·11-s − 0.192·12-s + 0.624·13-s + 0.908·14-s − 0.115·15-s + 0.250·16-s + 0.772·17-s + 0.602·18-s + 1.08·19-s + 0.149·20-s + 0.493·21-s + 0.568·22-s + 0.208·23-s + 0.135·24-s − 0.910·25-s − 0.441·26-s + 0.711·27-s − 0.642·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 + 0.665T + 3T^{2} \) |
| 5 | \( 1 - 0.670T + 5T^{2} \) |
| 7 | \( 1 + 3.39T + 7T^{2} \) |
| 11 | \( 1 + 2.66T + 11T^{2} \) |
| 13 | \( 1 - 2.25T + 13T^{2} \) |
| 17 | \( 1 - 3.18T + 17T^{2} \) |
| 19 | \( 1 - 4.73T + 19T^{2} \) |
| 29 | \( 1 + 9.07T + 29T^{2} \) |
| 31 | \( 1 - 2.40T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 2.12T + 47T^{2} \) |
| 53 | \( 1 - 2.74T + 53T^{2} \) |
| 59 | \( 1 - 2.19T + 59T^{2} \) |
| 61 | \( 1 - 0.869T + 61T^{2} \) |
| 67 | \( 1 + 5.55T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 - 0.381T + 73T^{2} \) |
| 79 | \( 1 + 6.31T + 79T^{2} \) |
| 83 | \( 1 + 8.49T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 3.23T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.76541334177531282021311305449, −7.10852275073400326911689895400, −6.09028163465065179085006678613, −5.88963934044936981753082582354, −5.16495485050462166271230042744, −3.76919340758210988829741513474, −3.09336593746800491337126625862, −2.37318711458875102805469288779, −1.01967887789016260037552235534, 0,
1.01967887789016260037552235534, 2.37318711458875102805469288779, 3.09336593746800491337126625862, 3.76919340758210988829741513474, 5.16495485050462166271230042744, 5.88963934044936981753082582354, 6.09028163465065179085006678613, 7.10852275073400326911689895400, 7.76541334177531282021311305449